Sommaire:

Supersymmetric extensions of conformal gravity have been known for a long time. In four spacetime dimensions, N=4 conformal supergravity is the maximally supersymmetric theory of this type. While the underlying multiplet of fields and their full non-linear transformations rules were already derived 40 years ago, no complete invariant action had been constructed so far. I will present the most general class of actions which turns out to be characterized by a holomorphic function of the scalar fields. This deviates from the non-maximally supersymmetric cases for which there is a unique action. I will then discuss how these results can be used to study certain higher-derivative couplings of N=4 (gauged) Poincaré supergravities.

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